ar X iv : 0 80 4 . 22 03 v 1 [ m at h . N A ] 1 4 A pr 2 00 8 REFINEMENT EQUATIONS AND SPLINE FUNCTIONS
نویسنده
چکیده
In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial C∞ solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis.
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